Based On Geometric Continuity Of Curves And Surfaces On The Extension Study

Parameter polynomial form is used in CAD,CAGD system and free curve surface expression widely. Such as Bernstein-Be zier, Schoenberg-B-Spline and Hermite-Coons. Bezier curve is one of the most basic and the most important shape-formatting tools in CAGD, it has wide applied background. The Bezier curve and surface are used widely especially.In the curves and surfaces design application, because the express ability of a single piece of parametric curve or surface is finite, if the shape of the curve and surface is complex, the segmental technology must be used to denote the curve or the surface, user can get the more complex curve or surface through several pieces of curves or surfaces combined fairly. A question usually coming is to extend a piece of curve or surface to a known point or a piece of curve, the extended curve or surface is also presented in parametric curve or surface with the same degree, and at the joint of the two curves or two pieces of surfaces arrive certain smoothness. Through this method several pieces of curve can be combined together, so the more complex shape of curve or surface can be expressed. The method is use for parametric curve and surface extend usually arrive parametric continuity. If at the joint between parametric curves and surfaces arrive the highest continuity, the curves or surfaces is unadjustable, so can't get the most fairly curve or surface.Because geometric continuity can fit the need of the smoothness between curves and surfaces, aim at the problem presented above, a new method of parametric curve and surface extended is presented in this paper. Geometric continuity is used to describe the smoothness of two pieces of parametric curves and surfaces at their joint in this paper, which offers two additional freedom degrees for adjusting the shape of the extended curve, thus the unadjustability of cubic curves in parametric continuity is removed. The shortest arc-length, minimum energy and minimum curvature variation of the curve, are used to set up the objective functions, respectively. The freedom degrees of the extended curve are determined by minimizing the objective functions,so can get the more fairing curves or surfaces, and analyze the existence of the freedom degrees. The new method is calculated simply, the extended curve and surface is more fairing, and have less energy and more smooth curvature variation. User can adjust the curves and surfaces through modify the freedom degrees, or combine two objective functions as the new objective function to get the freedom degrees, the facility of curves and surfaces adjusted is enhanced. But the three objective functions cannot always get the best result, besides, under some cases cannot get the freedom degrees, this is a problem we will research in the future.Because the objective functions are confirmed with approximate shortest arc-length, minimum energy and minimum curvature variation formula, the freedom degrees confirmed with the objective function is not the optimal result of the extended curve and surface. The precise energy and precise curvature variation formula is used as the objective functions to confirm the extended curve. The extended curve have more ideal shape and fairness.